Asymptotically optimal Boolean functions

Abstract

The largest Hamming distance between a Boolean function in n variables and the set of all affine Boolean functions in n variables is known as the covering radius ρn of the [2n,n+1] Reed–Muller code. This number determines how well Boolean functions can be approximated by linear Boolean functions. We prove thatlimn→∞⁡2n/2−ρn/2n/2−1=1, which resolves a conjecture due to Patterson and Wiedemann from 1983.